[PPT] - Weighted dependency graphs Valentin Fray Institut fr Mathematik, PowerPoint Presentation

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Weighted dependency graphs

Valentin Féray

Institut für Mathematik, Universität Zürich

Final conference of the MADACA project Domaine de Chalès, June 20th – June 24th 2016

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SLIDE 2

Central limit theorems

Theorem If Y1, Y2, . . . are independent identically distributed variables with finite variance, and Xn = n

i=1 Yi, then Xn−E(Xn) √Var Xn d

→ N(0, 1). (CLT)

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Central limit theorems

Theorem If Y1, Y2, . . . are independent identically distributed variables with finite variance, and Xn = n

i=1 Yi, then Xn−E(Xn) √Var Xn d

→ N(0, 1). (CLT) Relax identical distribution hypothesis − → Lindeberg condition.

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Central limit theorems

Theorem If Y1, Y2, . . . are independent identically distributed variables with finite variance, and Xn = n

i=1 Yi, then Xn−E(Xn) √Var Xn d

→ N(0, 1). (CLT) Relax identical distribution hypothesis − → Lindeberg condition. Relax independence hypothesis: leads to CLT for Markov chains, martingales, mixing sequences, exchangeable pairs, determinantal point processes, dependency graphs, . . .

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Central limit theorems

Theorem If Y1, Y2, . . . are independent identically distributed variables with finite variance, and Xn = n

i=1 Yi, then Xn−E(Xn) √Var Xn d

→ N(0, 1). (CLT) Relax identical distribution hypothesis − → Lindeberg condition. Relax independence hypothesis: leads to CLT for Markov chains, martingales, mixing sequences, exchangeable pairs, determinantal point processes, dependency graphs, . . . Goal of the talk: give an extension of dependency graphs that has a wide range of application.

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Dependency graphs

(Petrovskaya/Leontovich, Janson, Baldi/Rinott, Mikhailov, 80’s)

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Dependency graphs

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each edge {i, j} is taken independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2

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Dependency graphs

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each edge {i, j} is taken independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2 Question Fix p ∈ (0; 1). Does the number of triangles Tn satisfy a CLT?

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Dependency graphs

A problem in random graphs

Erdős-Rényi model of random graphs G(n, p): G has n vertices labelled 1,. . . ,n; each edge {i, j} is taken independently with probability p; 1 2 3 4 5 6 7 8 Example : n = 8, p = 1/2 Question Fix p ∈ (0; 1). Does the number of triangles Tn satisfy a CLT? Tn =

∆={i,j,k}⊂[n]

Y∆, where Y∆(G) =

1

if G contains the triangle ∆;

therwise.

Tn is a sum of mostly independent variables.

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Dependency graphs

Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family {Yα, α ∈ A} if if A1 and A2 are disconnected subsets in L, then {Yα, α ∈ A1} and {Yα, α ∈ A2} are independent. Roughly: there is an edge between pairs of dependent random variables.

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Dependency graphs

Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family {Yα, α ∈ A} if if A1 and A2 are disconnected subsets in L, then {Yα, α ∈ A1} and {Yα, α ∈ A2} are independent. Roughly: there is an edge between pairs of dependent random variables. Example Consider G = G(n, p). Let A = {∆ ∈ [n]

3

} (set of potential triangles) and

{∆1, ∆2} ∈ EL iff ∆1 and ∆2 share an edge in G. Then L is a dependency graph for the family {Y∆, ∆ ∈ [n]

3

}.
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Dependency graphs

Definition (Petrovskaya and Leontovich, 1982, Janson, 1988) A graph L with vertex set A is a dependency graph for the family {Yα, α ∈ A} if if A1 and A2 are disconnected subsets in L, then {Yα, α ∈ A1} and {Yα, α ∈ A2} are independent. Roughly: there is an edge between pairs of dependent random variables. Example Consider G = G(n, p). Let A = {∆ ∈ [n]

3

} (set of potential triangles) and

{∆1, ∆2} ∈ EL iff ∆1 and ∆2 share an edge in G. Then L is a dependency graph for the family {Y∆, ∆ ∈ [n]

3

}.

✞ ✝ ☎ ✆

Note: L has degree O(n)

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Dependency graphs

Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

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Dependency graphs

Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Theorem (Janson, 1988) Assume that

Nn

∆n

1/s ∆n

σn → 0 for some integer s. Then Xn satisfies a CLT.

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Dependency graphs

Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Theorem (Janson, 1988) Assume that

Nn

∆n

1/s ∆n

σn → 0 for some integer s. Then Xn satisfies a CLT.

For triangles, Nn = n

3

, ∆n = O(n), while σn ≍ n2. (for fixed p)

Corollary Fix p in (0, 1). Then Tn satisfies a CLT. (also true for pn → 0 with npn → ∞; originally proved by Rucinski, 1988).

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Dependency graphs

Applications of dependency graphs to CLT results

mathematical modelization of cell populations (Petrovskaya, Leontovich, 82); subgraph counts in random graphs (Janson, Baldi, Rinott, Penrose, 88, 89, 95, 03); Geometric probability (Avram, Bertsimas, Penrose, Yukich, Bárány, Vu, 93, 05 , 07); pattern occurrences in random permutations (Bóna, Janson, Hitchenko, Nakamura, Zeilberger, 07, 09, 14). m-dependence (Hoeffding, Robbins, 53, . . . ; now widely used in statistics) is a special case. (Some of these applications use variants of Janson’s normality criterion, which are more technical to state and omitted here. . . )

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Dependency graphs

Not an application of dependency graphs

Random graph G(n, M): G has n vertices labelled 1,. . . ,n; The edge-set of G is taken uniformly among all possible edge-sets of cardinality M. Example with n = 8 and M = 14 1 2 3 4 5 6 7 8

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Dependency graphs

Not an application of dependency graphs

Random graph G(n, M): G has n vertices labelled 1,. . . ,n; The edge-set of G is taken uniformly among all possible edge-sets of cardinality M. Example with n = 8 and M = 14 1 2 3 4 5 6 7 8 If p = M/ n

2

, each edge appears with probability p, but no independence

any more!

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Dependency graphs

Not an application of dependency graphs

Random graph G(n, M): G has n vertices labelled 1,. . . ,n; The edge-set of G is taken uniformly among all possible edge-sets of cardinality M. Example with n = 8 and M = 14 1 2 3 4 5 6 7 8 If p = M/ n

2

, each edge appears with probability p, but no independence

any more! Question Fix p ∈ (0; 1) and M = p n

2

. Does the number of triangles Tn in

G(n, Mn) satisfy a CLT?

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Dependency graphs

Not an application of dependency graphs

Random graph G(n, M): G has n vertices labelled 1,. . . ,n; The edge-set of G is taken uniformly among all possible edge-sets of cardinality M. Example with n = 8 and M = 14 1 2 3 4 5 6 7 8 If p = M/ n

2

, each edge appears with probability p, but no independence

any more! Question Fix p ∈ (0; 1) and M = p n

2

. Does the number of triangles Tn in

G(n, Mn) satisfy a CLT? Tn still writes as a sum of Y∆, but the Y∆ are pairwise dependent!

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Dependency graphs

Solution: edge-weighted dependency graphs

weighted graphs = graphs with weights in [0, 1] on edges. Definition A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if (skipped for the moment). Roughly: the smaller the weight on the edge {Yα, Yβ} is, the closer to independence Yα and Yβ should be.

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Dependency graphs

Solution: edge-weighted dependency graphs

weighted graphs = graphs with weights in [0, 1] on edges. Definition A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if (skipped for the moment). Roughly: the smaller the weight on the edge {Yα, Yβ} is, the closer to independence Yα and Yβ should be. Example Consider G = G(n, M), where M = p n

2

. Let A = {∆ ∈

[n]

3

} and

wt

L({∆1, ∆2}) =

1

if ∆1 and ∆2 share an edge in G. 1/n2

therwise.

Then L is a weighted dependency graph for the family {Y∆, ∆ ∈ [n]

3

}.
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Dependency graphs

Solution: edge-weighted dependency graphs

weighted graphs = graphs with weights in [0, 1] on edges. Definition A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if (skipped for the moment). Roughly: the smaller the weight on the edge {Yα, Yβ} is, the closer to independence Yα and Yβ should be. Example Consider G = G(n, M), where M = p n

2

. Let A = {∆ ∈

[n]

3

} and

wt

L({∆1, ∆2}) =

1

if ∆1 and ∆2 share an edge in G. 1/n2

therwise.

Then L is a weighted dependency graph for the family {Y∆, ∆ ∈ [n]

3

}.

✎ ✍ ☞ ✌

Note: L has degree O(n3), but weighted degree O(n).

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Dependency graphs

A normality criterion for weighted dependency graphs

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a weighted dependency graph Ln with weighted maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

V. Féray

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Dependency graphs

A normality criterion for weighted dependency graphs

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a weighted dependency graph Ln with weighted maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Theorem (F., 2016) Assume that

Nn

∆n

1/s ∆n

σn → 0 for some integer s. Then Xn satisfies a CLT.

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Dependency graphs

A normality criterion for weighted dependency graphs

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a weighted dependency graph Ln with weighted maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Theorem (F., 2016) Assume that

Nn

∆n

1/s ∆n

σn → 0 for some integer s. Then Xn satisfies a CLT.

For triangles in G(n, Mn), Nn = n

3

, ∆n = O(n), while σn ≍ n3/2.

Corollary Fix p in (0, 1) and set Mn = p n

2

. Then Tn satisfies a CLT.

(also true for n ≪ Mn ≪ n2; originally proved by Janson, 1994).

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Dependency graphs

Applications of weighted dependency graphs

crossings in random pair-partitions; subgraph counts in G(n, M); random permutations; particles in symmetric simple exclusion process; subword counts in Markov chains; patterns in multiset permutations, in set-partitions; spins in Ising model*; determinantal point process**. *in progress with Jehanne Dousse and Marko Thiel. **project (Some of these applications use a variant of the above normality criterion, which is more technical to state. . . )

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Cumulants

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Cumulants

What are (mixed) cumulants?

The r-th mixed cumulant κr of r random variables is a specific r-linear symmetric polynomial in joint moments. Examples: κ1(X) :=E(X), κ2(X, Y ) := Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) := E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z). Notation: κℓ(X) := κℓ(X, . . . , X).

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Cumulants

What are (mixed) cumulants?

The r-th mixed cumulant κr of r random variables is a specific r-linear symmetric polynomial in joint moments. Examples: κ1(X) :=E(X), κ2(X, Y ) := Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) := E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z). Notation: κℓ(X) := κℓ(X, . . . , X). If a set of variables can be split in two mutually independent sets, then its mixed cumulant vanishes.

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Cumulants

What are (mixed) cumulants?

The r-th mixed cumulant κr of r random variables is a specific r-linear symmetric polynomial in joint moments. Examples: κ1(X) :=E(X), κ2(X, Y ) := Cov(X, Y ) = E(XY ) − E(X)E(Y ) κ3(X, Y , Z) := E(XYZ) − E(XY )E(Z) − E(XZ)E(Y ) − E(YZ)E(X) + 2E(X)E(Y )E(Z). Notation: κℓ(X) := κℓ(X, . . . , X). If a set of variables can be split in two mutually independent sets, then its mixed cumulant vanishes. If, for each r big enough, we have κr(Xn) = o(Var(Xn)r/2), then Xn satisfies a CLT. (Janson, 1988)

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Cumulants

Sketch of proof of Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

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Cumulants

Sketch of proof of Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Fix r ≥ 1. Then κr(Xn) =

i1,...,ir

κ(Yn,i1, · · · , Yn,ir ).

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Cumulants

Sketch of proof of Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Fix r ≥ 1. Then κr(Xn) =

i1,...,ir

κ(Yn,i1, · · · , Yn,ir ). Each summand is 0, unless, up to reordering, each ij is a neighbour in Ln of either i1,. . . ,ij−1.

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Cumulants

Sketch of proof of Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Fix r ≥ 1. Then κr(Xn) =

i1,...,ir

κ(Yn,i1, · · · , Yn,ir ). Each summand is 0, unless, up to reordering, each ij is a neighbour in Ln of either i1,. . . ,ij−1. → only (r!)2 Nn ∆r−1

n

non-zero terms, each of which is easily bounded by a constant.

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Cumulants

Sketch of proof of Janson’s normality criterion

Setting: for each n, {Yn,i, 1 ≤ i ≤ Nn} is a family of bounded random variables; |Yn,i| < M a.s. we have a dependency graph Ln with maximal degree ∆n − 1. we set Xn = Nn

i=1 Yn,i and σ2 n = Var(Xn).

Fix r ≥ 1. Then κr(Xn) =

i1,...,ir

κ(Yn,i1, · · · , Yn,ir ). Each summand is 0, unless, up to reordering, each ij is a neighbour in Ln of either i1,. . . ,ij−1. → only (r!)2 Nn ∆r−1

n

non-zero terms, each of which is easily bounded by a constant. |κr(Xn)| ≤ CrNn ∆r−1

n

.

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Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants.

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SLIDE 38

Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants. Definition (F., 2016) A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if, for any α1, . . . , αr in A,

κ(Yα1, · · · , Yαr )
≤ Cr M
L[α1, · · · , αr]
.
V. Féray

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SLIDE 39

Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants. Definition (F., 2016) A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if, for any α1, . . . , αr in A,

κ(Yα1, · · · , Yαr )
≤ Cr M
L[α1, · · · , αr]
.
L
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SLIDE 40

Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants. Definition (F., 2016) A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if, for any α1, . . . , αr in A,

κ(Yα1, · · · , Yαr )
≤ Cr M
L[α1, · · · , αr]
.
L[α1, · · · , αr]: graph induced

by L on vertices α1, · · · , αr.

α1 α2 α3 α4 ε2 ε3 1 ε ε

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SLIDE 41

Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants. Definition (F., 2016) A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if, for any α1, . . . , αr in A,

κ(Yα1, · · · , Yαr )
≤ Cr M
L[α1, · · · , αr]
.
L[α1, · · · , αr]: graph induced

by L on vertices α1, · · · , αr. M

K
: Maximum weight of a

spanning tree of K. In the example, M

L[α1, · · · , α4]
= ε2.

α1 α2 α3 α4 ε2 ε3 1 ε ε

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SLIDE 42

Cumulants

Back to weighted dependency graphs

→ what we should require in the definition of weighted dependency graphs is a bound on mixed cumulants. Definition (F., 2016) A weighted graph L with vertex set A is a weighted dependency graph for the family {Yα, α ∈ A} if, for any α1, . . . , αr in A,

κ(Yα1, · · · , Yαr )
≤ Cr M
L[α1, · · · , αr]
.

This is a simplified version of the definition; some of the applications need a more general but more technical version.

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Cumulants

On the normality criterion for weighted dependency graphs

Proof is rather easy, similar as Janson’s. (The reordering is given by Prim’s minimum spanning tree algorithm.)

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SLIDE 44

Cumulants

On the normality criterion for weighted dependency graphs

Proof is rather easy, similar as Janson’s. (The reordering is given by Prim’s minimum spanning tree algorithm.) Question How to prove that something is a weighted dependency graph for a family {Yα, α ∈ A}? i.e. prove a bound on every cumulant κ(Yα1, · · · , Yαr ). A priori not easy!

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SLIDE 45

Cumulants

On the normality criterion for weighted dependency graphs

Proof is rather easy, similar as Janson’s. (The reordering is given by Prim’s minimum spanning tree algorithm.) Question How to prove that something is a weighted dependency graph for a family {Yα, α ∈ A}? i.e. prove a bound on every cumulant κ(Yα1, · · · , Yαr ). A priori not easy! → in the next section, we give 3 general tools for that.

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SLIDE 46

Finding weighted dependency graphs

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SLIDE 47

Finding weighted dependency graphs

Stability by powers

Setting: Let {Yα, α ∈ A} be r.v. with weighted dependency graph L; fix an integer m ≥ 2; for a multiset B = {α1, · · · , αm} of elements of A, denote YB := Yα1 · · · Yαm.

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Finding weighted dependency graphs

Stability by powers

Setting: Let {Yα, α ∈ A} be r.v. with weighted dependency graph L; fix an integer m ≥ 2; for a multiset B = {α1, · · · , αm} of elements of A, denote YB := Yα1 · · · Yαm. Tool 1 The set of r.v. {YB} has a weighted dependency graph Lm, where wt

Lm(YB, YB′) =

max

α∈B,α′∈B′ wt L(Yα, Yα′).

In short: if we have a dependency graph for some variables Yα, we have also one for monomials in the Yα.

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Finding weighted dependency graphs

Back to triangles (1/2)

It is enough to find a weighted dependency graph for edge indicators: Ye = 1e∈G (e ∈ [n] 2

).

(Indeed, triangle indicators are product of edge indicators.)

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SLIDE 50

Finding weighted dependency graphs

Back to triangles (1/2)

It is enough to find a weighted dependency graph for edge indicators: Ye = 1e∈G (e ∈ [n] 2

).

(Indeed, triangle indicators are product of edge indicators.) We need to bound cumulants of the shape κ

Ye1, · · · , Yer
.

(1) A priori, there can be repetitions in the sequence e1, · · · , er.

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SLIDE 51

Finding weighted dependency graphs

Back to triangles (1/2)

It is enough to find a weighted dependency graph for edge indicators: Ye = 1e∈G (e ∈ [n] 2

).

(Indeed, triangle indicators are product of edge indicators.) We need to bound cumulants of the shape κ

Ye1, · · · , Yer
.

(1) A priori, there can be repetitions in the sequence e1, · · · , er. Tool 2 (informal version) In (1), we can replace repeated variables or variables linked by edges of weight 1 by their products → with Bernoulli variables, we can forget repetitions.

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SLIDE 52

Finding weighted dependency graphs

Back to triangles (2/2)

We only need to prove κ

Ye1, · · · , Yer
= O(n−2(r−1)),

for distinct edges.

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SLIDE 53

Finding weighted dependency graphs

Back to triangles (2/2)

We only need to prove κ

Ye1, · · · , Yer
= O(n−2(r−1)),

for distinct edges. Joint moments are explicit: let En = n

2

,

Jℓ := E(Ye1 . . . Yeℓ) = En−ℓ

Mn−ℓ

En

Mn

= (En − ℓ)!Mn!

En!(Mn − ℓ)!. Example: for r = 3, we need to prove J3 − 3 J2 J1 + 2J3

1 = O(n−4).

For fixed r, easy to check with a computer algebra system, but not easy to prove for general r.

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SLIDE 54

Finding weighted dependency graphs

The multiplicative criterion

Tool 3 (for edges in G(n, M)) The bounds κ

Ye1, · · · , Yer
= O(n−2(r−1))

(for all r ≥ 1) are equivalent to

r

i=1

J

−(r

i)

i

= 1 + O(n−2(r−1))

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SLIDE 55

Finding weighted dependency graphs

The multiplicative criterion

Tool 3 (for edges in G(n, M)) The bounds κ

Ye1, · · · , Yer
= O(n−2(r−1))

(for all r ≥ 1) are equivalent to

r

i=1

J

−(r

i)

i

= 1 + O(n−2(r−1)) Second statement is much easier to handle: it is “multiplicative” in Ji: can be done separately for each factorial factor. lots of cancellations in LHS. → form here, quite easy to prove that we have a weighted dependency

graph. . .
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SLIDE 56

Other dependency graphs

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SLIDE 57

Other dependency graphs

Uniform random permutations

Let σ be a uniform random permutation of size n. Set Y(i,s) = 1σ(i)=s. Joint moment for distinct i1, · · · , ir, s1, · · · , sr: E

Yi1,s1 · · · Yir,sr
= (n − r)!

n! .

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SLIDE 58

Other dependency graphs

Uniform random permutations

Let σ be a uniform random permutation of size n. Set Y(i,s) = 1σ(i)=s. Joint moment for distinct i1, · · · , ir, s1, · · · , sr: E

Yi1,s1 · · · Yir,sr
= (n − r)!

n! . It is a quotient of factorial factor, so it satisfies the multiplicity criterion. Thus we have a weighted dependency graphs for the Yi,s; and, therefore, also for monomials in Yi,s. No need to do any computation here.

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SLIDE 59

Other dependency graphs

Uniform random permutations

Let σ be a uniform random permutation of size n. Set Y(i,s) = 1σ(i)=s. Joint moment for distinct i1, · · · , ir, s1, · · · , sr: E

Yi1,s1 · · · Yir,sr
= (n − r)!

n! . It is a quotient of factorial factor, so it satisfies the multiplicity criterion. Thus we have a weighted dependency graphs for the Yi,s; and, therefore, also for monomials in Yi,s. No need to do any computation here. → gives a bivariate extension of a functional CLT of Janson and Barbour.

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SLIDE 60

Other dependency graphs

Markov chains

Setting: Let (Mi)i≥0 be an irreducible aperiodic Markov chain on a finite space state S; Assume M0 is distributed with the stationary distribution π; Set Yi,s = 1Mi=s.

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SLIDE 61

Other dependency graphs

Markov chains

Setting: Let (Mi)i≥0 be an irreducible aperiodic Markov chain on a finite space state S; Assume M0 is distributed with the stationary distribution π; Set Yi,s = 1Mi=s. Proposition We have a weighted dependency graph L with wt

L

{Yi,s, Yj,t}
= |λ2|j−i,

where λ2 is the second eigenvalue of the transition matrix. → CLT for linear statistics N

i=1 f (Mi) = i,s f (s)Yi,s.

Already known (huge literature on the subject).

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SLIDE 62

Other dependency graphs

Markov chains

Setting: Let (Mi)i≥0 be an irreducible aperiodic Markov chain on a finite space state S; Assume M0 is distributed with the stationary distribution π; Set Yi,s = 1Mi=s. Proposition We have a weighted dependency graph L with wt

L

{Yi,s, Yj,t}
= |λ2|j−i,

where λ2 is the second eigenvalue of the transition matrix. Corollary (using the stability by product) We have a weighted dependency graph Lm for monomials Yi1,s1, . . . , Yim,sm. → gives a CLT for the number of copies of a given word in (Mi)0≤i≤N. (Answers a question of Bourdon and Vallée.)

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SLIDE 63

Other dependency graphs

Discrete determinantal point processes

Setting: S discrete state space; X random subset of S. Definition X is a discrete determinantal point process (DPP) with kernel K if for any distinct s1, . . . , sr in S, P({s1, . . . , sr} ⊆ X) = E r

i=1

1si∈X

= det
K(si, sj)
1≤i,j≤r

.

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SLIDE 64

Other dependency graphs

Discrete determinantal point processes

Setting: S discrete state space; X random subset of S. Definition X is a discrete determinantal point process (DPP) with kernel K if for any distinct s1, . . . , sr in S, P({s1, . . . , sr} ⊆ X) = E r

i=1

1si∈X

= det
K(si, sj)
1≤i,j≤r

. Strange definition (not even clear a priori if such a process exists at all), but there are lots of example: random Young diagrams, taken with Poissonized Plancherel measure; mid-time positions of non-intersecting random walks conditioned to come back to their starting positions. eigenvalues of random matrices (continuous DPP);

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SLIDE 65

Other dependency graphs

Discrete determinantal point processes

Setting: S discrete state space; X random subset of S. Definition X is a discrete determinantal point process (DPP) with kernel K if for any distinct s1, . . . , sr in S, P({s1, . . . , sr} ⊆ X) = E r

i=1

1si∈X

= det
K(si, sj)
1≤i,j≤r

. Lemma (Soshnikov, 2000) If X is a discrete determinantal point process with kernel K, then, for any distinct s1, . . . , sr in S, κ(1s1∈X, . . . , 1sr ∈X) =

σ ε(σ) i K(si, sσ(i)),

where the sum runs over cyclic permutation in Sr.

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SLIDE 66

Other dependency graphs

Discrete determinantal point processes

Setting: S discrete state space; X random subset of S. Definition X is a discrete determinantal point process (DPP) with kernel K if for any distinct s1, . . . , sr in S, P({s1, . . . , sr} ⊆ X) = E r

i=1

1si∈X

= det
K(si, sj)
1≤i,j≤r

. Soshnikov cumulant formula ⇒ for each DPP, we have a weighted dependency graph with weights K(si, sj). again, CLT for linear statistic is known; Project: investigate CLT for “multilinear” statistics.

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SLIDE 67

Conclusion

We provide a general tool to prove CLT for sums of weakly dependent random variables. Other examples (d-regular graphs)? Can we prove other type of results: speed of convergence, large deviations, . . . ? (with P.-L. Méliot and A. Nikeghbali, we have such results for usual dependency graphs.) A theory of continuous weighted dependency graphs (to handle continuous determinantal point processes)?

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